Investigation of Coriolis Mixing in Benzene Citation Runnels, John Hulett (1983) Investigation of Coriolis Mixing in Benzene. Master's thesis, California Institute of Technology. doi:10.7907/yxqj-6804. https://resolver.caltech.edu/CaltechTHESIS:03292010-084636927 Abstract An explanation for some recent observations in the electronic absorption spectrum of benzene by Schlag and coworkers is proposed. Several vibrational lines of the S_0 to S_1 transition have been studied by these authors with the Doppler—free, two—photon technique developed earlier. The resolution of this technique is so high that individual Δ J=0, Δ K=0 rotational lines in the Q—branch of a given vibrational transition may be seen in the electronic absorption spectrum. A major progression in the two—photon spectrum is the 14^1_0 1^n_0 progression. It was found that the 14^1_0 1^0_0 and 14^1_0 1^1_0 transitions had well—resolved rotationalstructure, but the 14^1_0 1^2_0 transition had reduced, not well resolved rotational structure--except for the prominent K=0 lines for small to medium values of J. This suggested to us Coriolis mixing. The model presented here utilizes a suggestion of Riedle et al.The Coriolis coupling to neighboring vibrational states is followed by a rapid radiationless depletion of these states, consistent with the decreased fluorescence yield observed by Wunsch et al. Estimates are made of the density of states of the various symmetries; these estimates are then used to estimate the extent of the relevant Coriolis mixing. The model results in a large decrease in intensity of non—zero K lines relative to the K=0 lines for the 14^1_0 1^2_0 transition, but not for the 14^1_0 1^0_0 and 14^1_0 1^1_0 transitions, in agreement with the experimental observations, making a few assumptions discussed below, but without the use of adjustable parameters. Predictions are then made for the rotational structure of other strong transitions in the two—photon spectrum of benzene. Item Type: Thesis (Master's thesis) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Marcus, Rudolph A. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1983 Record Number: CaltechTHESIS:03292010-084636927 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03292010-084636927 DOI: 10.7907/yxqj-6804 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5640 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 15 Apr 2010 20:43 Last Modified: 07 Aug 2025 18:26 Thesis Files Preview PDF - Final Version See Usage Policy. 736kB Repository Staff Only: item control page

Investigation of Coriolis Mixing in Benzene Thesis by John Hulett Runnels In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California 1983 To the Memory of My Mother iii Acknowledgments I would like to thank my parents for their guidance and support over the years; my research advisor, Professor R. A. Marcus, for his understanding, insight, and generosity with his time; fellow group members Paul Siders and Bob Cave for many fruitful discussions; and the Fannie and John Hertz Foundation for their generous support of my graduate studies. iv Abstract An explanation for some recent observations in the electronic absorption spectrum of benzene by Schlag and coworkers? is proposed. Several vibrational lines of the S, to S, transition have been studied by these authors with the Doppler-free, two-photon technique developed earlier?., The resolution of this technique is so high that individual AJ=0, AK=0 rotational lines in the Q-branch of ae given vibrational transition may be seen in the electronic absorption spectrum. A major progression in the two-photon spectrum is the 141° progression?, It was found that the 1,0 and 14241 transitions had well-resolved rotational 14519 oto structure, but the 14513 transition had reduced, not well resolved rotational structure~-except for the prominent K=0 lines for small to medium values of J. This suggested to us Coriolis mixing. The model presented here utilizes a suggestion of Riedle et alt.The Coriolis coupling to neighboring vibrational states is followed by a rapid radiationless depletion of these states, consistent with the decreased fluorescence yield observed by Wunsch et al‘. Estimates are made of the density of states of the various symmetries; these estimates are then used to estimate the extent of the relevant Coriolis mixing. The model results in a large decrease in -intensity of non-zero XK lines relative to the K=0 lines for the 14512 transition, but not for the 14510 and 14515 transitions, in agreement with the experimental observations, making a few assumptions discussed below, but without the use of adjustable parameters. Predictions are then made for the rotational structure of other strong transitions in the two-photon spectrum of benzene. Table of Contents Introduction 1 Methods 5 Results and Discussion 8 Conclusions 27 Appendix 29 References 35 Investigation of Coriolis Mixing in Benzene Introduction The field of two-photon molecular spectroscopy has been a fruitful one in recent years’. Many states forbidden to one-photon processes may be observed with two photons. In particular, most ground-state wavefunctions are totally symmetric. For molecules with a center of symmetry, this means a g state. Only transitions from g to u states are allowed with a single photon, but many g to g transitions may be observed with two photons. The symmetry of a given vibrational state can often be inferred from polarization-dependent two-photon spectra, even in a liquid or gas, provided that the time of interaction with the photons is short compared with the mean rotation time‘, Another edvantage of two-photon spectroscopy is that Doppler-free (or nearly Doppler-free) spectra can be taken of a sample in the gas _ phase at relatively high temperatures by putting two laser beams of equal (or nearly equal) frequencies at 180° to one another, thus making the net momentum transfer zero (or nearly zero) and eliminating the (first-order) Doppler effect. This permits higher resolution spectra than could otherwise be observed. Recently Riedle et al measured the first Doppler-free two-photon spectrum of a polyatomic molecule, that of benzene?. The resolution w/Aw achieved, 1.5x10', was high enough that individual rotational lines in the Q-branches of the vibrational transitions could be distinguished. The position of these lines depends not on the rotational constants of either the ground state or the excited state, but on the differences of the rotational constants between the two states, and the lines are thus closely spaced. (The difference in frequencies for the Q-branch S, to S, transitions of the J=0, K=0 and J=10, K=0 states is less than 1 cn” -) After this initial work was published, Schlag and coworkers studied? more closely the structure of the Q-branches of the 14512 transitions for n = 0O, 1, and 2. These are among the most prominent lines of the spectrum. The v mode is the symmetric C-C stretch, with 2 81s symmetry, and the v,, mode is the antisymmetric C-C stretch, with boa symmetry. The symmetry of the S, state 1in is Boy? so each of the 14915 vibronic transitions has symmetry Ale? and is strongly allowed with two photons. The frequencies of the peaks of these transitions relative to the electronic origin were reported as 1564, 2488, and 3414 om, respectively?. (The positions of the rotationless vibrational origins are slightly different; e.g., the fundamental is 1570 cm 2.) The Q-branches of the 1,0 and 14241 transitions showed well-resolved rotational 14515 oto structure, but the rotational structure of the Q-branch of the 14612 transition was reduced and not well resolved, except for the sharp K=0 line for each small to medium value of J. (For large J (around 20), even the K=0O lines disappeared, and a new, unidentified spectrum was seen instead.) The persistence of the K=0 lines when the others are reduced is suggestive of Coriolis coupling. Following a suggestion of Riedle et al+, we calculated the harmonic density of states of the upper surface of benzene as a function of the states’ symmetries, and found that it is reasonable to anticipate Coriolis coupling at the higher energy, but not the lower energies. With the ad hoc assumption that at least one of these Coriolis-coupled states is strongly coupled to a non-radiative manifold, one obtains a strong reduction in the intensity of the nonzero K lines. For example, if there is equal mixing of the initial state with one other state, the fluorescence from the initial level is now spread over two to four states (see below), which in turn are distributed over a 0.15 cm width (see also below). This effect is sufficient to reduce considerably the fluorescence intensity from those of lines of original width of 0.01 cm? to those of lines distributed over a region of width 0.15 om? or greater, depending on the value of K. The broadened nonzero K lines now largely form a "background" on which the K=0 lines stand out as silhouettes. Any substantial radiationless transition of the new states reduces this background by a factor of ten or greater (see below). At higher energies, lifetime broadening from non-radiative decay can reach the order of 0.1 cm 4, so the effect could occur with a lower density of states than would otherwise be required. Methods The perturbing Coriolis Hamiltonian for a symmetric top molecule is ‘—-_ H Popi/t, ’ where Ps is the component of the total angular momentum about the z-axis, Pr. is the component of the vibrational angular momentum about the z-axis, and I, is the moment of inertia about the z-axis. (The z-axis is the sixfold symmetry axis of the benzene molecule.) Coriolis coupling between degenerate states gives rise to a first-order splitting of the energy®, +2€AK, where € represents the Coriolis constant, and A=K7/21 We estimated the strength of the interaction between two nearly degenerate states by assuming an "effective Coriolis constant” of 0.8 (in line with the larger ones calculated by Robey and Schlag’), K=1 (since the intensity is decreased even for small K), and A=0.09 cm 12, This gives a Coriolis interaction energy of about 0.15 on !, We therefore assume Coriolis coupling, followed by coupling to a non-radiative manifold, occurs when the density of states of the appropriate symmetry reaches the order of six per wavenumber. In this case, the intensity of the non-zero K lines in the calculated spectrum was reduced by a factor of ten, since a reduction of the fluorescence excitation spectrum by at least such a factor was observed experimentally?. The rotational constants used to calculate the spectrum were taken from Riedie et al?. The original, unperturbed spectrum of the Q-branch is shown in figure 1. On" (T-W2) AONANDAYSA me) 00 I CO0° “¢ Ov oO'E Adbyll ayy “A 00 I a2" 00°1 08°0 09°0 on*d 020 00°0 oz "0- On*O- \ (Q39UNLY SUNN} 7 JYNLIAYLS 1 4undl4 HINHYE-8 Results and Discussion (a) Results of Riedle et al Two (harmonic) vibrations will undergo Coriolis coupling if the direct product of the symmetries contains the symmetry of the rotation involved*, In the case of benzene, which has point group Dew the relevant rotation is that about the figure axis, Ro which has symmetry lin The 14°71 states have symmetry bow and will therefore fog: couple with states of symmetry b since the direct product lu = 8 1u * bon Bo - The strength of this coupling depends on both the difference in energy of of these symmetries is b the two states, which we consider here, and on the Coriolis matrix element, which we have assumed to be that resulting from an "effective Coriolis constant” of 0.8 (see above). The total density of vibrational states as a funetion of energy is shown in figure 2. The scattered points are the result of direct counting, while the solid line is obtained from the Whitten-Rabinovitch approximation®,. The agreement between the two is remarkably close. (The frequencies for this calculation were taken from Ref. 7.) The total number of states of energy up to 3500 om” 4 was found to be about 80,000; this calculation took about 3 CPU seconds on a VAX 11/780. JO"OSE UO "UCcE 00°O08e L 00° Oe ;O1* (T-WO) AQONANDAYS 00 °00¢ e JdYNII4 00°09 00 °0<T G0 *OR 00°O 0 SALULS JO ALISNIQ WwlOl vd 00°O0R T go°G8 LISNSQ A ) 00021 J Ysd —W T G0 °00e i O0°Ofic 10 The next step was to calculate the density of states of a given symmetry as a function of energy. This was done by detailed counting on a digital computer, taking into account the symmetries of combination and overtone states?°’11, The results for each of the twelve symmetries of a Den molecule are shown in figures 3 through 14. Particular attention should be directed to figure 9, which shows the density of states for symmetry bia The densities are about 7, 1, and <1/2 states per om 2 for energies of 3414, 2488, and 1564 om +,respectively, corresponding to the 1414° states for n = 2, 1, and 0O, respectively. The first density is high enough to satisfy the criterion described above, around 6 states per om 7, especially when it is considered that anharmonic considerations will increase the density of states somewhat. The second and third densities are too low for there to be a high probability of coupling to the non-radiative manifold. With the ad hoc assumption that at least one state coupled to the 1471 state is strongly coupled to the non-radiative manifold, and that this coupling thereby reduces the intensity of nonzero K _ lines by a factor of ten, we get the perturbed spectrum shown in figure 15. The nonzero K lines would actually be spread out even further because of Coriolis splitting, and perhaps the +K degeneracy would be split by higher-order Coriolis interactions as well. In the absence of detailed calculations of these two splittings, we have omitted their 00 "OSE 00 °0c 00°O08e L Ox 00 ‘Oe 00 (T-WI) XIN age = 00031 4 NO0dds 00°0dT 00°08 00°Oh 00°) a] 11 € JYynolad S4dlu ITy AWYLAWWAS HLIM 1S dO ALISNSAG 00° ane /O1x (T-WO) ADNANDAYS J0° OSE O0°OcE 00 *08<e 00 °Ohe 00°00¢ 00°O091 00°O0c 00°08 00°Oh 00°05 _ ieiaial pealieiadinieiealiatl ~ _ . = fan) ° | nm a oIn on ro) “OTVAS oO cores —2 ‘ ne - oz : On -: a Lae —< - in . OlD on) - = | a | Ops et : "oo cS oO 924 PS on) AHLIWWAS HLIM oO fh Fynot- SAlIHLS 4O ALISNSAC OB "OSE 00 °0cE 00°*Che ;OTx (T-W2) AJNANDAdS go"o0e =—0aosl G0 °OcT 00°08 00 °Oh 00°O 13 00 °08¢ GS 4ynols ty Ad l SALBLS AWWAS HLIM 40 ALISNAOG 00 “¢ Q0 oo°n JALISNSAQ I 60'9 J ddd ;OT* (T-WI) AQNANOSdS covose = oo"oee =— 00082 =o ORe =D OBE =— 00091 00°02! 00°08 oo"On olenae) m0) ee _ -— mame ae -_ « ia) - ee ee es — wm eee ome CO ott De TRTLIILT ~< 14 ( Ney AYLAWWAS HLIM 9 4YNII4 SdLbLS 40 ALISNIAQ ,OI* (P-WO) JONANDAYS oo"o9e = gd ace oo7ose = oo"Ohe = 00002 ~———00" 091 00°02] 00°08 00°Oh 00° 2. LST ISIS TT et wo 7 oO Me lam suaneoemmee coumeees oad : mrt |r wee lee cee “o Oo Ops ca IT g ~ uo AHLSIWWAS HLIM oe) L 3ynota S4LbLS Ab ALISNAO IN co ae) ;OTx CIT-WI) AINSANDAYS oovoge = 0*oee =— a OBe = OH OHe = BN*OO2 ~=— 00091 00*021 00°08 o0*On Clanas L JIN In turt. ovr? a —a oe ones ~ » swe mee .- = Cy \o . ed an AYLAWWAS HLIM 8 dynos S3LbLS 40 ALISNSAC ao ane) a uo GO "OSE O0°0ce 00 °O08e 00° “OLX Ofe 17 (T-WO) AONAND AYA 00°O00c ae I G0 °O0cT 00°08 00 °OR 00°O wees o . ee ee seeee . ee ee . . . ae ase = . see = mene 8 . - «= oe os ee ee ee 6 qun9l a Alg AYLIWWAS HLIM S4LviS 40 ALISNSAG 50°C T-WI OO °O9E 18 00 °0ce 00°08e¢ 00 ‘Ofc O0°O00c 00°09 ,OIx = (E-WO) AUNANDAYS l L O0°O0cT 00°08 00 °OR 00°70 O01 3xYNSIS ee ee . aes amass uses a 06k ee - 8 com em im emi - - . - . . . we me =o. . es ee . - . . <_< . mee ee me ewe - meee en ome ooo - ee ee . «8 = 8 see eee _— —_e . ee rr - . . smn se ome . a aoee . oe . a ewe = as . . —s we eee om mee Neg AYLAWWAS HLIM SALYLS 40 ALISNAG 00°C OW%e ‘OT Oo ho mane) ca a) O0°O9E U0 °Oce ;OT™* (T-W2) AONANDAYSA O00°00¢ 19 Tt Sdynola o0°O9T D0°OctT 00°08 00 °OR 00° 913 AYLAWWAS HLIM SdL¥LS 40 ALISN3O 00 1 “OT 00 00 “WO dad) ALISNSO CO ‘fe Ox (1 -WO) AONANDAYS 20 el 923 Lal dynos S4dLyLlS JWWiS HLIM 40 ALISNSAO oo'oSs ~=«©6roooe~g)~=——Sioe2 =O OMe~=—soo002~—— 0009! ~— 0001 00°08 00‘Oh oo "o, ; : L—. vee exert Son leer r le = ri = ns “ id rt BYRRLP VRS UE: “3 | | oO Mm (oe) “3 Olx (1-W2) LONINOSYS 00 "OSE O0°O0cE 00°08¢ 00 °Ohe 00°00e¢ O0°O9T O0°O0ct 00°08 00 °On 00°C a0) AY o0 "8 DO 1 dad) LLISN4AOQ | c 00° T “fe 0 S| Woe mene, 21 oe) ATs AYIAWWAS HLIM fan) El 3YnNoI4 SS3lblS 40 ALISNAG O1x (1-WQ) LONSNO SYS G0 °O9e O0*OcE 00 °08¢ ite Q0°00e¢ O0°O9T 00 *Ocl 00°08 ae eee a eee . = 4 vs 00 °OR 00°O 22 Al 4aynola Neg AWLISWWAS HLIM SJLULS 40 ALISNSO 00 "3] JL LISNAG Ud) c ooh “WI dae On* 23 (T-WO) AONANDAGSA Qe" oo'T 08°0 09°O0 On’ 0 0e°0 o0°O Oc *O- Oh'O- L L L LL L i ! 1 . ia) | = ia) | 3 (a) 1D me) ws) oy oD OD | u | <x |_ (or) O un (O439¥Nlyad) 2 JHA LINYLS : ST SaYunolg HINHYE-O [a OO 24 contribution to the spectrum shown. For comparison, the experimental spectra of ref. 1 are shown in figure 16. (Note the different energy scales.) The qualitative agreement with the calculated spectra of figures 1 and 15 is good. (b) Predictions for Other Transitions One prediction our model makes is that if this phenomenon were observed in other transitions with weaker Coriolis coupling, there might be some rotational structure for low K even when that for larger K is obscured, since the strength of the Coriolis interaction is proportional to K. Another prediction this model can make is whether this type of rotational structure is to be expected in other transitions, assuming the primary state is not itself strongly coupled to a non-radiative manifold. Of the stronger two-photon transitions in benzene? with symmetry this structure might be observable in the 15132 Pou’ 0° 0 transition at 2995 on? , with a harmonic density of baa states around 3 per cm}, but would probably not be seen 2 -1 for the 145115 transition at 2599 cm , with a harmonic density around 1 per cm 25 0 ob- O2- oe POGUUUUTTG FOTURUTETG SRTORTOUG FTRTTETOTS oe- [2HO)A VY 07- os- 093- OL- o8- 06- yr os ae ae oe [as Ue Cae Cas Me ae Ome Os Se ee i on on Oe ae mrss raaala gy $sa xa p- WI ZITE = | iy i Mi \" 9T FundTd tue 2672 = SSaaKa5 0, 0 oat 999 (pd Ayisuaju] 82U99S9J0N} 4 Of the transitions with symmetry Cay which couple _ to other states of the same symmetry, the structure might be 142 iti 2770 cm 2 ith orto transition at cm > wi a harmonic density around 7 per cm i, but would probably not observable in the 18 be seen for the 1451664 transition occurring at 2391 cm? 2 with a harmonic density around 2 per emt, 26 27 Conclusions We have performed some calculations confirming the qualitative reasonableness of a suggestion by Riedle et al? to explain their recent observations of the rotational structure in the Q-branches of _ the 14518 vibrational progression in the S, to S, excitation spectrum of benzene. 1 We have made predictions regarding the likely rotational structure of several transitions in this spectrum. In brief, we concluded that anomalous rotational structure is likely when the density of states with the proper symmetry for Coriolis coupling to the primary state is of the order of six per wavenumber. For weaker coupling, there might be a transition from unperturbed lines for small K to the perturbed spectrum for larger K. The plots for density of states of the different symmetries as a function of energy appear similar to that for the total density of states, apart from a scale factor. (Each of the eight nondegenerate symmetries has about 4 percent of the total number of states, and each of the four degenerate symmetries has about 17 percent.) It would be interesting to see whether this is in fact the case at high energies, and if so, whether some statistical description of these densities could be found analogous to the Whitten-Rabinovitch approximation. 28 One aspect of the observed spectrum that the present study does not explain is the onset of a new structure in 1,2 the 145 0 band at high values of J. One possibility is that for higher J (stronger perturbations) there is a new zeroth~order Hamiltonian, Analogies occur in spectra with double wells as the coupling between the wells increases, and in the pendulum problem as the barrier to hindered rotation decreases. It wonld be interesting to investigate this possibility by studying the diagonalization of a large matrix of the relevant basis states, and examining the eigenvalue sequences, 29 Appendix The material in this appendix is concerned with some cubic anharmonic calculations on the upper surface of benzene which were done earlier in connection with the subject of this thesis, but which were not used in the -results reported here. This material is included for the possible future reference of someone doing anharmonic calculations on benzene, The force field used for the *Boy excited state was that of Ref. 1. The benzene molecule was assumed to be planar with D symmetry, although there is recent evidence 6h for a slight distortion from planarity in the excited state. The normal modes and frequencies were calculated using Wilson's F and G matrix method??, We used the simplest anharmonic perturbations to this harmonic model--the contributions to the potential energy from the diagonal cubic terms in the C-C and C-H bond lengths. The perturbing Hamiltonian is H’=) k, (ACC)? + ) k,(ACH)? (It has been observed that the diagonal terms in bond lengths are the most important contributions to both cubic and quartic anharmonic potentials+3.) We initially used the cubic force constants calculated by Pulay et al*‘, but found that these were too large by a factor of about four 30 to be consistent with the 15 overtones of the observed spectrum of Garforth and Ingold?*, who found good agreement with the anharmonic form E = khw(n + 1/2) - xhow(n + 1/2)? » where kw = 924 om and xkw = 0.5 om 2, Instead we used an anharmonicity consistent with these overtones. (This is discussed further below.) First-order nondegenerate perturbation theory was used to calculate the anharmonic vibrational wavefunctions, This calculation is consistent with our use of second-order perturbation theory to calculate the energies of some of the vibrational overtones, (An odd perturbation leaves’ the energies unchanged to first order.) Initially we used the cubic force constants of Pulay et ali‘ to calculate the anharmonic perturbations in benzene, but obtained perturbations that were far too large. (These constants were -—34.0 aJ/A3 for the C-H o stretches and -40.7 aJ/A? for the C-C_ stretches.) As a Dn check, we calculated the overtone frequencies for the 1p (symmetric C-C stretch) progression, which are known to deviate only slightly from a purely harmonic progression?’, The calculated frequencies were too anharmonic. New values for both constants of —9.0 aJ/At gave reasonable agreement with the first few terms of the progression?®, (These results are relatively insensitive to the anharmonicity of 31 the C-H stretches since the principal motions involved are C-C stretches.) There are several possibilities for the origin of the discrepancy with the calculation of Pulay et al. The published values were for the ground state, not the excited state, but it seems unlikely that this would cause a difference of half an order of magnitude since the ground 0 state symmetric c-C stretch progression 1 in the phosphorescence spectrum of benzene is also quite harmonic?7’, The reported ab initio values were apparently calculated from the distortions of individual bonds, not symmetric (or antisymmetric) combinations of bond distortions. This could make a significant difference in the higher order corrections (which are, after all, relatively small). (It should be noted that the same anharmonic corrections do not necessarily apply to all distortions; rather, they are a function of the normal coordinate(s) involved.) Finally, it might be the case that constants is not as accurate for higher order terms. The cubic contributions to the wavefunctions were in the neighborhood of 1 to 8 percent. The F and G matrix method of Wilson?? is a time-honored method of finding frequencies and normal modes of polyatomic molecules. (Chapter 10 is devoted to a 32 sample calculation for benzene.) There are standard routines for doing this, but doing it oneself allows for more flexibility and makes calculating Coriolis constants fairly simple since one has the B matrix needed for the Coriolis constants as an intermediate result from generating the G matrix. ("Dummy"” zero frequency modes may have to be appended to the B matrix to make it square before inversion if the zero frequency modes are initially separated out of the coordinate system.) The EISPACK eigensystem package of the NATS (National Activity to Test Software) served very well in diagonalizing the matrices, and the Givens procedure did well inverting matrices, Although we found fault with their anharmonic constants, the harmonic force field of Pulay et al for the ground state of benzene14 seems to be a good one if needed. (We actually did not use the ground state surface anywhere in these calculations since we did not explicitly compute Franck-Condon factors.) Robey and Schlag's force field for the excited state of benzene’? seems to be the best to date. One should pay close attention to their sign conventions, which are a little confusing, but are defined in the body of the paper. One should also note that different authors use different internal coordinates or have what appear to be the same coordinates when in fact some differ by multiplicative constants. We performed the calculations in internal coordinates rather than symmetry coordinates since internal coordinates are more versatile for anharmonic 33 calculations or for extensions to substituted benzenes; the amount of CPU time needed to diagonalize a 30x30 matrix with EISPACK is negligible. There are two conventions for labeling the normal modes of benzene. The one that appears to be more common, and the one used here, is that of Pitzer and Scotti&, (The other is that of Herzberg*®.) In the harmonic approximation for a planar molecule, the in-plane modes and out-of-plane modes separate exactly. For a particular calculation, it may be sufficient to calculate just the in-plane modes. (In the calculations reported here, the only information needed regarding the out-of-plane modes was their frequencies, which were added in an ad hoc manner.) However, there is evidence that benzene in the *Bo a state is slightly puckered?, It might be interesting to take the puckering into account; however, no dramatic changes in the Q-branch rotational lines would be expected since the molecule would still be a symmetric top. To calculate the anharmonic wavefunctions, standard first-order nondegenerate perturbation theory was used. (Second-order theory was used to find the positions of the overtones; these were used to estimate the cubic constants.) Diagonalization of the matrix GF gives the matrix L, where S=LQ, S represents the internal coordinates, and Q@ the normal coordinates. The cubes of the bond length displacements may then be written as a 34 third order polynomial in the Q's, and the resulting integrals for the matrix elements are separable and may be performed analytically. (We found them with the aid of MACSYMA, the MIT symbolic manipulation language.) The development of anharmonic force fields is not nearly so extensive as that of harmonic force fields, especially for polyatomic molecules. Plivat® might serve as ai useful starting point, but the literature is sparse. Estimating anharmonicity constants from overtone and combination progressions might be as good a method as any. 35 References 1. E. Riedle, H. J. Neusser, and E. W. Schlag, J. Phys. Chem 86, 4847 (1982). 2. E. Riedle, H. J. Meusser, and E. W. Schlag, J. Chem. Phys. 75, 4231 (1981). 3. L. Wunsch, H. J. Neusser, and E. W. Schlag, J. Chem. Phys. 66, 386 (1977). 4. L. Wunsch, H. J. Neusser, and E. W. Schlag, Z. Naturforsch. 36A, 1340 (1981). 5. W. M. McClain, Acct. Chem. Res. 7, 129 (1974). 6. E. B. Wilson, J. C. Decius, and PP. C. Cross, Molecular Vibrations, McGraw-Hill 1955, p. 367. 7. M. J. Robey and E. W. Schlag, J. Chem. Phys. 2775 (1977). IN ™ J 8. G. Herzberg, Infrared and Raman Spectra, Van Nostrand 1945, p. 126. 9. P. J. Robinson and K. A. Holbrook, Unimolecular Reactions, Wiley-Interscience 1972, p. 131. 10. Wilson, Decius, and Cross, p. 332. 11. Herzberg, p. 127. 12. Wilson, Decius, and Cross, chap. 4. 13. J. Pliva in Critical Evaluation of Chemical and Physical Structural Information, D. R. Lide and M. A. Paul (ed.), National Academy of Sciences 1974, p 289. 36 14. P. Pulay, G. Fogarasi, and J. E. Boggs, J. Chem. wh Phys. 74, 3999 (1981). 15. F. M. Garforth and C. K. Ingold, J. Chem. Soc., 417 (1948). 16. The deviations from harmonicity expected from the results of Garforth and Ingold® in th 1™ states for n= 1, 2, and 3 are 1.0, 3.0, and 6.0 cm", respegtively. The values which were obtgined for k,=k,= -9.0 aJ/A*® were 1.6, 3.5, and 5.5 cm ~, respectively. 17. E. R. Bernstein, S. D. Colson, D. S. Tinti, and G. W. Robinson, J. Chem. Phys. 48, 4632 (1968). 18. K. S. Pitzer and D. W. Scott, J. Am. Chem. Soc., 803 (1943). 19. Herzberg, p. 118.